Suppose we wanted to prove something similar but simpler, namely that any colouring must include three equally-spaced points in a line all the same colour. How would we go about it?
First, note that we can certainly find two points the same colour! Pick two, and draw a line through them, marking them as "-1" and "1". This gives a scale on which you can label other points on the line, too. The points we are interested in are -3, 0 and 3.
It's easy to check that however you colour these three points, you will get three equally-spaced points of one colour. You can try it out yourself below.
Incidentally, sharp readers might have noticed that the gizmo above is very similar to the one in this issue's article on Ramsey theory.